# Framework for Flexural Rigidity Estimation in Euler-Bernoulli Beams Using Deformation Influence Lines

^{*}

## Abstract

**:**

## 1. Introduction

- sensitivity to impairment,
- foundation in mechanical theory,
- consistency in the evaluation technique, and
- efficacy and efficiency of the technique in practice.

## 2. Influence Lines for Euler-Bernoulli Beam Evaluation

## 3. Flexural Rigidity Estimation (FRE)

#### 3.1. Derivation for FRE

#### 3.2. Alternate Derivation for FRE

## 4. Calculation of Moment $\mathit{m}\left(\mathit{\xi}\right)$

#### 4.1. Statically Determinate Systems

#### 4.2. Statically Indeterminate Systems

#### 4.2.1. Deflection Influence Line (DIL)

#### 4.2.2. Rotation Influence Line (RIL)

## 5. Analytical Illustrations of the Method

#### 5.1. Example I—Statically Determinate System

#### 5.2. Example II—Statically Indeterminate Systems

#### 5.2.1. Using Deflection Influence Line (DIL)

#### 5.2.2. Using Rotation Influence Line (RIL)

## 6. Application

#### 6.1. Algorithm for Addressing Noisy Measurements

- It will become zero at the location of each support.
- The value of the function and its first derivative would become equal at both the left- and right-hand boundaries of the considered pieces.

**lsqlin**function implemented in the MATLAB programming software. The constraints and the objective function are defined as linear equations in the format ${\mathrm{A}}_{\mathrm{cts}}\mathrm{X}={\mathrm{b}}_{\mathrm{cts}}$ and ${\mathrm{A}}_{\mathrm{obj}}\mathrm{X}={\mathrm{b}}_{\mathrm{obj}}$, respectively, where vector $\mathrm{X}$ is the vector of unknowns. Appendix A presents the method for calculating these matrices. Resultant matrices ${A}_{obj},\text{}{b}_{obj},\text{}{A}_{cts},\text{}$ and $\text{}{b}_{cts}$ are input into the

**lsqlin**function using the “Interior-Point” algorithm [32].

#### 6.2. Application on a Numerical Model

#### 6.3. Application on a Real World System

## 7. Summary

## Author Contributions

## Conflicts of Interest

## Appendix A

## References

- American Society of Civil Engineers (ASCE). Report Card for America’s Infrastructure; American Society of Civil Engineers (ASCE): Reston, VA, USA, 2017; Available online: http://www.infrastructurereportcard.org (accessed on 1 February 2017).
- Rytter, A. Vibration Based Inspection of Civil Engineering Structures. Ph.D. Thesis, Aalborg University, Aalborg, Denmark, 1993. [Google Scholar]
- Doebling, S.W.; Farrar, C.R.; Prime, M.B. A Summary Review of Vibration-Based Damage Identification Methods; Technical Report LA-UR-98-0375; Los Alamos National Laboratory: Los Alamos, NM, USA, 1997.
- Sohn, H.; Farrar, C.R.; Hemez, F.; Czarnecki, J. A Review of Structural Health Monitoring Literature: 1996–2001; Technical Report LA-13976-MS; Los Alamos National Laboratory: Los Alamos, NM, USA, 2001.
- Worden, K.; Dulieu-Barton, J.M. An overview of intelligent fault detection in systems and structures. Int. J. Struct. Health Monit.
**2004**, 3, 85–98. [Google Scholar] [CrossRef] - Fan, W.; Qiao, P. Vibration-based damage Identification methods: A review and comparative study. Struct. Health Monit.
**2011**, 10, 83–92. [Google Scholar] [CrossRef] - Story, B. Structural Impairment Detection Using Arrays of Competitive Artificial Neural Networks. Ph.D. Thesis, University of Texas A&M, College Station, TX, USA, 2012. [Google Scholar]
- Seo, J.; Hu, J.; Lee, J. Summary review of structural health monitoring applications for highway bridges. J. Perform. Constr. Facil.
**2015**, 30. [Google Scholar] [CrossRef] - Das, S.; Saha, P.; Patro, S.K. Vibration-based damage detection techniques used for health monitoring of structures: A review. J. Civ. Struct. Health Monit.
**2016**, 6, 477–507. [Google Scholar] [CrossRef] - Turer, A. Conditional Evaluation and Load Rating of Steel Stringer Highway Bridges Using Field Calibrated 2D-Grid and 3D-FE Models. Ph.D. Thesis, University of Cincinnati, Cincinnati, OH, USA, 2000. [Google Scholar]
- Zaurin, R.; Catbas, F.N. Integration of computer imaging and sensor data for structural health monitoring of bridges. Smart Mater. Struct.
**2010**, 19. [Google Scholar] [CrossRef] - Zaurin, R.; Catbas, F.N. Structural health monitoring using video stream, influence lines, and statistical analysis. Struct. Health Monit.
**2010**, 10, 309–332. [Google Scholar] [CrossRef] - Story, B.A.; Fry, G.T. Methodology for designing diagnostic data streams for use in a structural impairment detection system. J. Bridge Eng.
**2013**, 19. [Google Scholar] [CrossRef] - Story, B.A.; Fry, G.T. A structural impairment detection system using competitive arrays of artificial neural networks. Comput. Aided Civ. Infrastruct. Eng.
**2014**, 29, 180–190. [Google Scholar] [CrossRef] - Zeinali, Y.; Story, B.A. Structural impairment detection using deep counter propagation neural networks. J. Procedia Eng.
**2016**, 145, 868–875. [Google Scholar] [CrossRef] - Sładek, J.; Ostrowska, K.; Kohut, P.; Holak, K.; Gaska, A.; Uhl, T. Development of a vision based deflection measurement system and its accuracy assessment. Measurement
**2013**, 46, 1237–1249. [Google Scholar] [CrossRef] - Oh, B.K.; Hwang, J.W.; Kim, Y.; Cho, T.; Park, H.S. Vision-based system identification technique for building structures using a motion capture system. J. Sound Vib.
**2015**, 356, 72–85. [Google Scholar] [CrossRef] - Feng, D.; Feng, M.Q. Experimental validation of cost-effective vision-based structural health monitoring. Mech. Syst. Signal Process.
**2017**, 88, 199–211. [Google Scholar] [CrossRef] - Catbas, F.N.; Zaurin, R.; Susoy, M.; Gul, M. Integrative Information System Design for Florida Department of Transportation: A Framework for Structural Health Monitoring of Movable Bridges; Final Report BD548-11; Florida Department of Transportation: Tallahassee, FL, USA, 2007. [Google Scholar]
- Chen, Z.; Zhu, S.; Xu, Y.; Li, Q.; Cai, Q. Damage detection in long suspension bridges using stress influence lines. J. Bridge Eng.
**2014**, 20. [Google Scholar] [CrossRef] - Turer, A.; Levi, A.; Aktan, A.E. Instrumentation Proof-Testing and Monitoring of Three Reinforced Concrete Deck-on-Steel Girder Bridges Prior to, During and after Superload; University of Cincinnati Infrastructure Institute: Cincinnati, OH, USA, 1998. [Google Scholar]
- Catbas, F.N.; Zaurin, R.; Gul, M.; Gokce, H. Sensor networks, computer imaging, and unit influence lines for structural health monitoring: Case study for bridge load rating. J. Bridge Eng.
**2012**, 17, 662–670. [Google Scholar] [CrossRef] - Zaurin, R.; Khuc, T.; Catbas, F. Hybrid sensor-camera monitoring for damage detection: Case study of a real bridge. J. Bridge Eng.
**2016**, 21. [Google Scholar] [CrossRef] - Bernal, D. Damage localization and quantification from the image of changes in flexibility. J. Eng. Mech.
**2014**, 140, 279–286. [Google Scholar] [CrossRef] - Catbas, F.N.; Lenett, M.; Aktan, A.E.; Brown, D.L.; Helmicki, A.J.; Hunt, V. Damage Detection and Condition Assessment of Seymour Bridge; Proceedings of the SPIE Series; Society of Photo-Optical Instrumentation Engineers: Bellingham, WA, USA, 1998; pp. 1694–1702. [Google Scholar]
- Stimac, I. Influence of sampling interval on deflection-influence-line-based damage detection in beams. In Proceedings of the 5th International Conference on Civil Engineering-Science and Practice, Žabljak, Montenegro, 17–21 February 2014; pp. 355–361. [Google Scholar]
- Stimac, I.; Grandić, D.; Bjelanović, A. Comparison of techniques for damage identification based on influence line approach. Mach. Technol. Mater.
**2011**, 7, 9–13. [Google Scholar] - Stimac, I.; Mihanović, A.; Kožar, I. Damage detection from analysis of displacement influence lines. In Proceedings of the International Conference on Bridges (Structural Engineering Conferences), Dubrovnik, Croatia, 21–24 May 2006; pp. 1001–1008. [Google Scholar]
- Wang, C.Y.; Huang, C.K.; Chen, C.S. Damage assessment of beam by a quasi-static moving vehicular load. Adv. Adapt. Data Anal.
**2011**, 3, 417–445. [Google Scholar] [CrossRef] - Wang, Y.; Liu, X. Beam damage localization method considering random uncertainty using mid-span displacement data. In Proceedings of the Sustainable Development of Critical Infrastructure, Shanghai, China, 16–18 May 2014; pp. 438–446. [Google Scholar] [CrossRef]
- Li, X.Y.; Law, S.S. Adaptive Tikhonov regularization for damage detection based on nonlinear model updating. J. Mech. Syst. Signal Process.
**2010**, 24, 1646–1664. [Google Scholar] [CrossRef] - Mathworks. Global Optimization Toolbox: User’s Guide (r2017b). 2017. Available online: http://www.mathworks.com/help/pdf_doc/gads/gads_tb.pdf (accessed on 30 November 2017).
- Zeinali, Y.; Li, Y.; Rajan, D.; Story, B.A. Accurate structural dynamic response monitoring of multiple structures using one CCD camera and a novel targets configuration. In Proceedings of the 11th International Workshop on Structural Health Monitoring, Palo Alto, CA, USA, 12–14 September 2017; pp. 3107–3114. [Google Scholar]

**Figure 1.**Initial position and deflected shape of a beam structure under the effects of a unit load located at (

**a**) $\xi $; (

**b**) ${x}_{A}$.

**Figure 2.**Deflected shape of the modified beam with a replaced hinge at measurement point under the effects of a virtual displacement.

**Figure 4.**Example-I analysis results (

**a**) recorded Deflection Influence Line (DIL) (

**b**) first derivative of DIL (

**c**) second derivative of DIL (

**d**) beam flexural rigidity estimation (FRE).

**Figure 6.**Example-2 Analysis results (

**a**) recorded DIL; (

**b**) first derivative of DIL; (

**c**) second derivative of DIL; (

**d**) diagram of moment $m\left(\xi \right)$; and, (

**e**) beam flexural rigidity estimation (FRE).

**Figure 7.**Example-2 Analysis results; (

**a**) recorded RIL; (

**b**) first derivative of RIL; (

**c**) second derivative of RIL; (

**d**) diagram of moment $m\left(\xi \right);$ and, (

**e**) beam flexural rigidity estimation (FRE).

**Figure 9.**Results of application of the proposed method on a simply supported beam with polluted RILs.

**Figure 11.**Experiment results, (

**a**) measured RILs and fitted curves, (

**b**) Results of flexural rigidity estimation.

Parameter | Value | Unit |
---|---|---|

$E{I}_{0}$ | $10$ | $\mathrm{Force}/{\mathrm{Length}}^{2}$ |

$\beta $ | $0.6$ | - |

$L$ | $10$ | $\mathrm{Length}$ |

${x}_{0}$ | $2$ | $\mathrm{Length}$ |

${L}_{1}$ | $6$ | $\mathrm{Length}$ |

${L}_{2}$ | $7$ | $\mathrm{Length}$ |

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**MDPI and ACS Style**

Zeinali, Y.; Story, B.A.
Framework for Flexural Rigidity Estimation in Euler-Bernoulli Beams Using Deformation Influence Lines. *Infrastructures* **2017**, *2*, 23.
https://doi.org/10.3390/infrastructures2040023

**AMA Style**

Zeinali Y, Story BA.
Framework for Flexural Rigidity Estimation in Euler-Bernoulli Beams Using Deformation Influence Lines. *Infrastructures*. 2017; 2(4):23.
https://doi.org/10.3390/infrastructures2040023

**Chicago/Turabian Style**

Zeinali, Yasha, and Brett A. Story.
2017. "Framework for Flexural Rigidity Estimation in Euler-Bernoulli Beams Using Deformation Influence Lines" *Infrastructures* 2, no. 4: 23.
https://doi.org/10.3390/infrastructures2040023